## Estimating modal instability threshold for photonic crystal rod fiber amplifiers |

Optics Express, Vol. 21, Issue 13, pp. 15409-15417 (2013)

http://dx.doi.org/10.1364/OE.21.015409

Acrobat PDF (2511 KB)

### Abstract

We present a semi-analytic numerical model to estimate the transverse modal instability (TMI) threshold for photonic crystal rod amplifiers. The model includes thermally induced waveguide perturbations in the fiber cross section modeled with finite element simulations, and the relative intensity noise (RIN) of the seed laser, which seeds mode coupling between the fundamental and higher order mode. The TMI threshold is predicted to ~370 W – 440 W depending on RIN for the distributed modal filtering rod fiber.

© 2013 OSA

## 1. Introduction

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. **37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express **21**(2), 1944–1971 (2013). [CrossRef] [PubMed]

2. T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express **19**(8), 7398–7409 (2011). [CrossRef] [PubMed]

## 2. Thermally induced waveguide perturbation and mode distributions

10. E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol. **30**(22), 3494–3499 (2012). [CrossRef]

11. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” J. Quant. Electron. **37**(2), 207–217 (2001). [CrossRef]

*q*is the heat load

*Q*divided by the area of the active core

*A*.

_{core}*Q*is

*Q*in the core and 0 outside the core, and

_{0}*k*is the thermal conductivity of the

_{i}*i*’th layer. The parameters used in the calculations are given in Table 1.

*P*

_{pump}[12

12. M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power cw fiber lasers,” Proc. SPIE **7195**, 71951U, 71951U-11 (2009). [CrossRef]

*S*of the rod fiber is included as a reasonable approximation for determining the heat load as a function of signal power,

*P*

_{signal}.where

*α*is the pump absorption,

*λ*and

_{p}*λ*is the pump and signal wavelength. The fractiondescribes the change in pump power within

_{s}*dL*due to pump photons converted to signal photons, and the bracket is the quantum defect for the pump photons. The largest heat load is assumed at the fiber output end in a backwards pumped configuration, where the rod fiber attains the highest stimulated emission and thereby largest quantum defect heating. Hence the last 10 cm of the rod fiber

*dL*is considered for estimating the generated heat load as a function of signal power. The slope efficiency has been measured to 71% – 75% for the DMF85, and is set to 70% in the calculations. The thermal load, and therefore the threshold, for onset of TMI, depends on pump absorption. The small signal pump absorption is typically 20 dB/m for a rod fiber amplifier, but it decreases with increased population of the lasing level. The pump absorption is set to the realistic value of 10 dB/m in the calculations.

*T*is the temperature at the core edge, and

_{core}*r*is the core radius. The solution decays logarithmic outside the active material, where

_{core}*Q =*0 in Eq. (1).

*i*represents the current layer, thus

*T*is the temperature at the outer boundary, and

_{i}*r*is the outer radius of layer

_{i}*i*. The air cladding is thin silica bridges separating air holes, and is approximated by an effective thermal conductivity given by the number of air holes, the air clad width and the width of the silica bridges separating the air holes [13

13. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express **11**(22), 2982–2990 (2003). [CrossRef] [PubMed]

*T*

_{0}with forced convective cooling with coefficient

*h*, yielding a temperature at the fiber edge ofThe temperature increase of the fiber causes a refractive index increase across the entire fiber cross section due to the thermo-optic effect with coefficient

*η*.

14. Comsol, “Products,” <http://www.comsol.com/products/multiphysics/> (2 January 2013).

15. M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express **20**(7), 7263–7273 (2012). [CrossRef] [PubMed]

10. E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol. **30**(22), 3494–3499 (2012). [CrossRef]

^{−5}, smaller than technologically feasible, yielding an NA of 0.011 and

*V*= 2.40. The SIF core area is equal to the doped area of the DMF and PCF. The SIF SM properties suffer from increasing heat load causing the HOM core overlap to increase significantly making the fiber MM at

*Q*= 7 W/m, where the HOM content is larger than 50%. The DMF rod fiber experiences larger differential mode overlap within the SM regime compared to the SM SIF, indicating that the intentional HOM delocalization by the resonator elements in the cladding efficiently suppress HOM content in the core. The DMF85 rod fiber can tolerate a heat load up to

*Q*= 24 W/m, before the HOM content increases above 50%. This shows that the modal stability of this resonant structure is significantly more robust than in the SIF. The thermally induced waveguide perturbations also affect the mode field diameter (MFD) of the FM, since increasing thermal load yields larger core confinement, see Fig. 4, and is estimated to a linear decrease in MFD of 2% per 100 W of extracted output power for DMF85.

16. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express **20**(5), 5742–5753 (2012). [CrossRef] [PubMed]

## 3. Estimating modal instability threshold

*al.*[8

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. **37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express **21**(2), 1944–1971 (2013). [CrossRef] [PubMed]

*g*, and the transverse core overlap ratio Г of the FM and HOM determined from the mode calculations. The nonlinear coupling constant χ is given by [8

8. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. **37**(12), 2382–2384 (2012). [CrossRef] [PubMed]

9. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express **21**(2), 1944–1971 (2013). [CrossRef] [PubMed]

*η*is the thermo-optic coefficient and

*β*is the propagation constant.

*A*is an effective overlap integral that is determined from the mode distributions

*ψ*for the FM and HOMThe inner integral is over the doped core region, and the outer integral is over the full cross section for the transverse coordinates

*G*is the solution to the Fourier transformed transient heat equation that describes the change in temperature due to the thermal load

*Q*. Assuming

*G*to be translation invariant it is given bywhere

*K*

_{0}is the zeroth order modified Bessel function of second kind,

*ρ*is the density, and

*C*is the specific heat capacity.

*G*is a convolution operator that allows the inner integral in Eq. (8) to be determined by the convolution theorem.

*χ*in Eq. (7) is an odd function that tends towards zero as Ω approaches zero. Therefore, there must be a seed to initialize power transfer from the FM to the HOM. We consider TMI seeded by relative intensity noise (RIN)

*R*from the seed laser. The approximate expression for the fraction of HOM power content

_{N}*ξ*as a function of extracted average output power

_{out}*P*is given by [9

_{signal}**21**(2), 1944–1971 (2013). [CrossRef] [PubMed]

*P*is the seed power and Ω

_{seed}*is Ω for maximum*

_{p}*χ*, and

*ξ*is the fraction of power that is initially coupled to the HOM. A seed laser is characterized by its RIN, which can be on the order of −120 dBc/Hz to −100 dBc/Hz corresponding to a laser with low and high RIN.

_{in}*R*= −120 dBc/Hz and

_{N}*R*= −100 dBc/Hz, and represent a TMI threshold interval depending on RIN. The DMF85 rod fiber is initially SM at the signal wavelength with the SM window slightly blueshifting as the fiber heats up, and the HOM becomes guided at high thermal loads corresponding to high extracted output power. The thermal load is proportional to signal power, therefore the estimated TMI threshold occurs when the calculated TMI threshold matches the signal power used in the thermal load calculations. This happens at 371 W and 443 W, as indicated in Fig. 6. For comparison the TMI threshold of the PCF and theoretical SIF in Fig. 3(a) and Fig. 3(b) are estimated to 348 W – 426 W and 392 W – 470 W for

_{N}*R*= −100 dBc/Hz –

_{N}*R*= −120 dBc/Hz respectively. This is slightly higher for the PCF and slightly lower for the SIF compared to the DMF85, as expected based on the differential mode overlap in Fig. 3. The SM window of the DMF85 can be shifted by changing the designs parameters, which allow engineering the DMF85 rod fiber to have SM operation at high or low signal power depending on the application [15

_{N}15. M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express **20**(7), 7263–7273 (2012). [CrossRef] [PubMed]

16. M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express **20**(5), 5742–5753 (2012). [CrossRef] [PubMed]

6. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

19. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express **20**(22), 24545–24558 (2012). [CrossRef] [PubMed]

6. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express **20**(10), 11407–11422 (2012). [CrossRef] [PubMed]

## 4. Conclusion

## References and links

1. | D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B |

2. | T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express |

3. | F. Jansen, F. Stutzki, H.-J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express |

4. | A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express |

5. | T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express |

6. | B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express |

7. | C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express |

8. | K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. |

9. | K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express |

10. | E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol. |

11. | D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” J. Quant. Electron. |

12. | M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power cw fiber lasers,” Proc. SPIE |

13. | J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express |

14. | Comsol, “Products,” <http://www.comsol.com/products/multiphysics/> (2 January 2013). |

15. | M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express |

16. | M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express |

17. | F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. |

18. | H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express |

19. | A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(140.6810) Lasers and laser optics : Thermal effects

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 17, 2013

Revised Manuscript: June 13, 2013

Manuscript Accepted: June 13, 2013

Published: June 20, 2013

**Citation**

Mette Marie Johansen, Kristian Rymann Hansen, Marko Laurila, Thomas Tanggaard Alkeskjold, and Jesper Lægsgaard, "Estimating modal instability threshold for photonic crystal rod fiber amplifiers," Opt. Express **21**, 15409-15417 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15409

Sort: Year | Journal | Reset

### References

- D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B27(11), B63–B92 (2010). [CrossRef]
- T. T. Alkeskjold, M. Laurila, L. Scolari, and J. Broeng, “Single-mode ytterbium-doped large-mode-area photonic bandgap rod fiber amplifier,” Opt. Express19(8), 7398–7409 (2011). [CrossRef] [PubMed]
- F. Jansen, F. Stutzki, H.-J. Otto, M. Baumgartl, C. Jauregui, J. Limpert, and A. Tünnermann, “The influence of index-depressions in core-pumped Yb-doped large pitch fibers,” Opt. Express18(26), 26834–26842 (2010). [CrossRef] [PubMed]
- A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (2011). [CrossRef] [PubMed]
- T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express19(14), 13218–13224 (2011). [CrossRef] [PubMed]
- B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (2012). [CrossRef] [PubMed]
- C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express19(4), 3258–3271 (2011). [CrossRef] [PubMed]
- K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett.37(12), 2382–2384 (2012). [CrossRef] [PubMed]
- K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high-power fiber amplifiers,” Opt. Express21(2), 1944–1971 (2013). [CrossRef] [PubMed]
- E. Coscelli, F. Poli, T. T. Alkeskjold, M. M. Jørgensen, L. Leick, J. Broeng, A. Cucinotta, and S. Selleri, “Thermal Effects on the Single-Mode Regime of Distributed Modal Filtering Rod Fiber,” J. Lightwave Technol.30(22), 3494–3499 (2012). [CrossRef]
- D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” J. Quant. Electron.37(2), 207–217 (2001). [CrossRef]
- M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power cw fiber lasers,” Proc. SPIE7195, 71951U, 71951U-11 (2009). [CrossRef]
- J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express11(22), 2982–2990 (2003). [CrossRef] [PubMed]
- Comsol, “Products,” < http://www.comsol.com/products/multiphysics/ > (2 January 2013).
- M. M. Jørgensen, S. R. Petersen, M. Laurila, J. Lægsgaard, and T. T. Alkeskjold, “Optimizing single mode robustness of the distributed modal filtering rod fiber amplifier,” Opt. Express20(7), 7263–7273 (2012). [CrossRef] [PubMed]
- M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292W with improved mode stability,” Opt. Express20(5), 5742–5753 (2012). [CrossRef] [PubMed]
- F. Stutzki, H.-J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett.36(23), 4572–4574 (2011). [CrossRef] [PubMed]
- H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express20(14), 15710–15722 (2012). [CrossRef] [PubMed]
- A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express20(22), 24545–24558 (2012). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.